Lifting some approximation properties from a dual space $X’$ to the Banach space $X$

Abstract

For a fixed Banach operator ideal A, we characterize A-compact sets (in the sense of Carl and Stephani) that are determined by c0 via the Banach composition ideal A • K∞, with K∞ the Banach ideal of Fourie and Swart. This characterization allows us to relate KA-approximation properties on a Banach space and KB-approximation properties on its dual space, where A and B are ideals linked by some classical procedures. These approximation properties have been widely studied in several papers in the last years.

On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators

“…(3.3) Subsequently it was shown by Kim [22,Theorem 1.1] that (3.2) also holds for p = 1, and by Lassalle and Turco [28,Theorem 4.7] that also the claim (3.3) holds for p = 1. See also [26,Theorem 2.5] for a general result that includes both statements (3.2) and (3.3) for p = 1.…”

“…This paper is part of the Ph.D.-thesis of the author and he thanks his supervisor Hans-Olav Tylli for many valuable comments and remarks during the preparation of the manuscript. The author also thanks Pablo Turco for supplying reference [26] and for helpful comments.…”

“…The I-AP has recently been studied in several papers for various Banach operator ideals I, see e.g. [9,32,27,24,25,26] and their references. Moreover, results about special instances of the I-AP have been used in the recent studies [44,45] of radical quotient algebras of Banach operator ideals.…”

On approximation properties related to unconditionally p‐compact operators and Sinha–Karn p‐compact operators

“…In fact, the identities  𝑢𝑝 =   𝑝 ′ and  𝑝 =   𝑝 hold isometrically [30,31], where  𝑝 ′ and  𝑝 are the respective Banach operator ideals of the classical 𝑝 ′ -compact operators (𝑝 ′ is the dual exponent of 𝑝) and the right 𝑝-nuclear operators. The   -AP was introduced by Lassalle and Turco [30] and it has subsequently been studied for various classes of Banach operator ideals  in [26,28,31].…”

Maximal ideals of generalized summing linear operators

“…The theory of operator ideals is central in modern mathematical analysis (see [15], [28, 6.3]) and, in this context, maximal ideals play a key role. For recent developments on maximal operator ideals, see, e.g, [19,22,24,32]. A number of important operator ideals are defined, or characterized, by the transformation of vector-valued sequences; and some of these ideals are known to be maximal.…”